gebze institute of technology, school of business ...9)-1208-1219.pdf · asian economic and...

Asian Economic and Financial Review, 2014, 4(9): 1208-1219

1208

CREDIT PORTFOLIO SELECTION ACCORDING TO SECTORS IN RISKY

ENVIRONMENTS: MARKOWITZ PRACTICE

Halim Kazan

Gebze Institute of Technology, School of Business Adminstration ,Department of Operational Research,

Gebze- Kocaeli Turkey

Kültigin Uludağ

Central Bank of the Republic of Turkey-Ankara/Turkey

ABSTRACT

In this study, it was researched that how the rate of repayment of loans will be increased and how

the credit risk will be minimized in banking sector, by using Markowitz Portfolio Theory.

Construction, textile and wholesale and retail sectors were examined under the central bank data.

Portfolio groups were selected and risks( variances of Portfolio groups) were evaluated according

to Markowitz portfolio theory. Markowitz portfolio theory is effective than the other portfolio

selection instruments. Although Classical risk measurement tools measure risks, but they do not be

able to answer how the risks can be reduced. On the other hand, Markowitz portfolio model, which

is used in this study, show how the risks can be reduced.

© 2014 AESS Publications. All Rights Reserved.

Keywords: Markowitz portfolio theory, Credit portfolio selection, Performing loans.

1. INTRODUCTION

In banking sector, diversification of credits according to sectors are important to understand

repayment ratios of loans or credit risks in detail. Credit reports on the basis of sectors show the

financial performances of sectors periodically. This research is based on maximizing performing

loans of banks , besides reduce the volatility of repayment –risk- . Unfortunately, banks have

faced with credit defaults. Banks have faced with 3 problems that are caused by credit defaults; 1)

Repayment time of defaulted loans is unknown. 2) Defaulted loans may become non-performing

and it may be totally defict. 3) Legal period and costs of liquidation of default credits exceed the

principles.

Banks calculate the loss rate in case of default according to Basel II standards. This loss

increases the credit costs and also affects profit of banks in a negative way. In the light of

foregoing, high performing rate of credits provide banks to produce stable and forward-looking

Asian Economic and Financial Review

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policy. Moreover this also enables to reduce liquidity crunch which is results from decrease in

repayment of credits because of credit defaults. The credits provided by banks are followed from

Central Bank monthly data of non-accruing loans according to sectors. It is important to examine

performing loans according to distinct sectors because sectors are different in terms of their

internal dynamics. For instance, The credits which is used by sector A have better performance

than sector B according to performing loans however sector A can have high volatility repayment

performance.This shows in some case higher performing loans rate means higher risks. . The

mortgage crisis rose in USA at 2008 directly affected mining industry in Turkey since mining

sector‘s firms mostly export their goods to the USA.This crisis caused close down the most

mining firm. The other aspect of that crisis was performing loans of mining firms were sharply

decreased and defauld credits were increased respectively. In 2008 financial global crisis the

banks faced with huge losse. For instance a bank consider only sectors‘ with respect to performing

loans rate and disregard the volatility of the repayment performance than the bank may face with

unexpected credit repayment rate because of the risks-volatility of repayment rate-.Therefore it is

important for banks to minimize the volatility of repayment rate and maximize performing loans so

banks should determine a portfolio group of sectors to split the risk into different sectors and avoid

inherent risk of sectors as mining sector mentioned obove.

2. LITERATURE REVIEW

Optimizing a portfolio is a major area in finance. The aim of portfolio aptimization is

maximizing the income of portfolio, simultaneously minimizing the risk. One way of optimizing a

portfolio was suggested by Harry Max Markowitz (born August 24, 1927) who published the

article -Portfolio selection- in Journal of Finance 1952. The most important aspect of Markowitz‘

model was his description of the impact on portfolio diversification by the number of securities

within a portfolio and their covariance relationships (Megginson, 1996). His studies , Markowitz

(1952), were first published in The Journal of Finance. Subsequently, Markowitz significantly

expanded his studies in his book, Portfolio Selection: Efficient Markowitz (1959). In 1990 he

received the Nobel Memorial Prize in Economic Sciences due to his contributions to portfolio

theory.

In this study, our purpose is to how a bank extend credits to the sectors to aim for maximizing

performing loans . In order to understand this study we need some background information about

the concepts of portfolio theory. Modern Portfolio Theory is based upon Harry Max Markowitz‘s

studies. Before Markowitz, prices of stock exchanges had been determined by ―Theory of

Investment Value‖ which was stated by John (1938). According to Theory of Investment, it was

enough to regard expected value for determining future stoc value that is when an investor want to

maximize the profit it was enough looking into only one stock. However, Markowitz told that risk

should not be ignored in a risky environment. He associated risk with variance and came up with a

new theory Markowitz (1991). Markowitz analyzed stocks under the historical datas in a risky

environment and studied to minimize risks while maximize yields. In addition, Tomas et al. (2012)


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analyzes the evolution of the systematic risk of the banking industries in eight advanced countries

using weekly data from 1990 to 2012.

Radovan and Juraj examined o find determinants of Loss given default using a set of firm loan

micro-data of an anonymous Czech commercial bank. Antoine and Bertille (2012)develop a new

framework to provide a feasible optimal investment rule that accounts for estimation risk. Disatnik

and Katz (2012) introduce a portfolio strategy that generates a portfolio, with no short sale

positions, that can outperform the 1/N portfolio.Tu and Zhou (2011) propose an optimal

combination of the naive 1/N rule with one of the four sophisticated strategies the Markowitz rule,

the Jorion (1986) rule, the Mackinlay and Pastor (2000) rule, and the Kan and Zhou (2007) rule as

a way to improve performance. They found that the combined rules not only have a significant

impact in improving the sophisticated strategies, but also outperform the 1/N rule in most

scenarios. Graham (2013)examined systematically the contributes to the existing literature by

showing: (a) that the CAPM fails empirically when applied to industries; (b) that after 1993 the

value and momentum anomalies appear to continue whereas the beta anomaly appears to have

weakened; and (c) that the value and momentum anomalies, and the value of beta, can largely be

ignored if the task is to estimate industry cost of equity. Bai et al. (2009) develop new bootstrap-

corrected estimations for the optimal return and its asset allocation and prove that these bootstrap

corrected estimates are proportionally consistent with their theoretic counterparts.

Alexander and Gordon (2009) first introduced value-at-risk as a measure of risk and how it

relates to standard deviation, the risk measure at the heart of the model of Markowitz. Second,

similarly introduced conditional value-at-risk (also known as expected shortfall) as a measure of

risk and compare it with VAR.

Third, they briefly introduced stress testing as a supplemental means of controlling risk. Bris et

al. (2008), discussed how securities investors could protect themselves from risk through

diversification.Greyserman et al. (2006) studied portfolio selection methodology using a Bayesian

forecast of the distribution of returns by stochastic approximation.

Lee (1977), Litzenberger and Kraus (1976) made alternative studies about distribution of

return in portfolio theory (Elton et al., 1998). without regard to developed by William Sharpe, John

Lintner, and Jan Mossin, another important capital markets theory evolved as an outgrowth of

Markowitz‘ and Tobin‘s earlier works—The Capital Asset Pricing Model (CAPM) (Megginson,

1996). The CAPM provided an important evolutionary step in the theory of capital markets

equilibrium, better enabling investors to value securities as a function of systematic risk.

Sharpe (1964) significantly advanced the Efficient Frontier and Capital Market Line concepts

in his derivation of the CAPM. Sharpe would later win a Nobel Prize in Economics for his seminal

contributions. A year later, Lintner (1965) derived the CAPM from the perspective of a corporation

issuing shares of stock. Finally, in 1966, Mossin also independently derived the CAPM, explicitly

specifying quadratic utility functions (Megginson, 1996). Since the earlier works of Markowitz,

and later, Sharpe, Lintner and Mossin, there have been various expansions and iterations of MPT.

The remainder of this essay addresses a perceived ―simplicity‖ gap in that literature, and suggests a


1211

systemic failure of theorists and practitioners to capitalize upon the tremendous advances in finance

and technology. It also specifically extends the conceptual premises of Wharton professor, Benniga

(2006), wherein he argues for a more simplistic approach to understanding and calculating the

various mathematical concepts underlying MPT.

Overall, the risk component of MPT can be measured, using various mathematical

formulations, and reduced via the concept of diversification which aims to properly select a

weighted collection of investment assets that together exhibit lower risk factors than investment in

any individual asset or singular asset class. Diversification is, in fact, the core concept of MPT and

directly relies on the conventional wisdom of ―never putting all your eggs in one basket‖ (Fabozzi

et al., 2002; Veneeya, 2006; Mcclure, 2010). It is instructive to note here that Markowitz‘ portfolio

selection theory is a ‗normative theory.‘ Fabozzi et al. (2002) define a normative theory as ―one

that describes a standard or norm of behavior that investors should pursue in constructing a

portfolio. Conversely, Sharpe‘s asset pricing theory (CAPM) is regarded as a ‗positive theory‘—

one that hypothesizes how investors actually behave as opposed to how they should behave.

Together, they provide a theoretical framework for the identification and measurement of

investment risk and the development of relationships between expected return and risk. There

remains a degree of debate as to whether or not MPT is interdependent upon the validity of asset

pricing theory (Fabozzi et al., 2002). This analysis assumes that MPT is indeed independent of

asset pricing theory, with the latter concept the subject of separate analysis. Portfolio Theory

basically based upon selection of best portfolio group from listed stocks. Portfolio Theory

arguments could have been applied in different areas. For example; In agricultural economies, grain

prices are periodic and market price of grains are changed month by month.

Because there are more than one grain types and their harvest times are different..For this

reasen storage problems are risen for producers who have limited storage area what kind of grains

are stocked up and what their quantities should be are problems for warehouse owner . In addition

, unknown future market price of stored grains pose a risk. In that case , maximizing the return of

the portfolio consist of different varietis of stored grains under the risk –unknown future market

price- is a problem. That problem was illustrated using quadratic equation and solved under linear

constraint by using Markowitz-Portfolio arguments Heifner (1966). Todays there are legal football-

betting web sites which are based on to estimate score of matches.. Due to the scores which are

occured at a point of match or end of the match, bets‘ return changes Match of the soccer teams are

analyzed statistically using historical data. For this analyze it is accepted that the goals are

reasonable for constant mean- Poisson distribution. Times of goals are not certain according to the

probability so it entertains risk. In addition, mean of time intervals of goals may changes. In

football betting game return rate is a negative function of mean of scoring a goal in a time interval.

From this point of view, for different time intervals different bets can be played and different

returns can be gained. In this position while maximizing the return the risk will increase. Thus,

different forecasts for different time intervals will be selected for portfolio group. Appropriate

portfolio group which optimize the return is selected according to Markowitz Theory Fitt (2009).


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One of the main problem of a company is to construct its own supply-chain. Company should

determine its suppliers in terms of revenues and reliability of the suppliers. If the company can not

obtain the goods which are needed to continue its production, this may cause counterparty default

risk and also bankruptcy of the firm. Because a single supplier is a great risk , firms should work

with alternative suppliers and make a combination of suppliers (portfolio group) to maximize the

revenues which minimize risks. To sum up problem is formulated below;

N=number of suppliers

( ) ∑ ( ) ( 1)

Return on portfolio

∑

∑ ∑

( 2)

Variance of portfolio=Portfolio risk

Where

is variance and is corrolation coefficient

And the equation (2) is minimized by using Markowitz Theory arguments by Lao and Liu

(2007). The technology of generating of the electrical energy is important for energy efficiency and

production costs. In USA and Switzerland the technology of electrical energy (coal, gas, nuclear,

wind, oil, solar etc.) was changed in years according to production costs. The return was different

for different combination of sources. Krey and Zweifel (2008), in their studies, determined which

combinations provide the optimum return for USA and Switzerland by using Markowitz mean-

variance Portfolio Theory. Moreover, in Tunisia, in the study of energy generation planning for

2010-2020 the question of how the optimum portfolio from nuclear sources and fossil sources is

selected, was answered by using Markowitz Portfolio Theory. (Abdelhamid et al., 2009)

3. THE PURPOSE OF THE STUDY AND METHODOLOGY

The purpose of this research; banks gave credit to sectors which include that the banks of

credits at maximum viability rate so that they create portfolios.

In this context; The Central Bank data, t. C Wholesale, Retail Trade, Construction, textile and

textile products from the sectors of modern portfolio theory has been applied to the arguments of

credits separated by Markowitz.

Banks ' total loan volume ratio of importance since the viability of the study is to minimize the

risks and maximize the vitality of credit rates. Study; "Central Bank of the Republic of Turkey law

No. 1211 44 within the framework of the 3rd item by banks will be liquidated in cash and loans

obtained by the declarations" had used loans to sectors related to how much using data from live

(the rate of viability of loans).

A historical data calculated data is obtained on a monthly basis. The average portfolio within

the historical data to calculate the variance analysis of the risks of the sector, namely ® cast.

Volatility calculated this way, the risk of each sector with the variance.


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The expected value of the portfolio (vitality) ( ) ∑ ( ) ( )

constituent

element of a percentage of the value of each is the sum of the products of the sector in the form of

expected with. In theory, the variance for each sector or entering stock portfolio (Risk) is calculated

separately and then outlining the variance of a function of the portfolio.

Various elements that make up the portfolio of this function in probability (Xi) portfolio-what

is risk. This study of the reasons which have been used in the theory of proportions (Xi) Markowitz

portfolio elements be combined portfolio2

p (portfolio risk) minimum is calculated, the value of

the portfolio turnover (Xi) values instead of optimal portfolio and its goal has been reached

4. THEORY AND THE MODEL

In a time series which shows the performing rate of loans for each sector percent and if n is the

number of sectors in portfolio group;

is percent of sector in portfolio

is variance of the sector

is covariance of sectors

( )is expected value of sector

For the portfolio p which has n sectors, providing that is variance of portfolio and ( ) is

expected portfolio value;

Equation 1

( ) = ( )+ ( )+……….+ ( )

= (

)

+( )

+( )

+……

+…..

+( )

İf the equation 1 is rewritten

Equation 2

( ) ∑ ( )

= ∑

+ ∑ ∑

is get.

Because the aim of the study is minimize the risks and the variance of portfolio represents the

risk the model will be as follow:


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Model 1

Minimize [ = (

)

+( )

+( )

+

+…..

+( ) ]

Constraint 1:

[In the model are percent of each component in portfolio. In other words, summation of is

equals to 100% of portfolio. Summation of probability of has to equal to 1]

Constraint 2: , , ,

[In the model the aim is providing credits absolutely for each sector so the percentage of each

sector should be greater than 0. In other words >0]

If the Model 1 is rewritten in more complex form;

Model 2

Minimize = ∑

+ ∑ ∑

∑ =1 2) >0 i= 1,2……..n will be get.

The optimization problem is solved under constraints.

5) Application and solution of the problem

Z= Construction

Y= Textile

X= Wholesale and Retail

If ,

and are variances of sectors respectively, , ve are covariances between

sectors;

= ,

= ,

=4696 ,

=

=

=

When the values are put into the model

( )= +

+

Variance of portfolio X,Y and Z will be as shown at (1).

With quadratic programming variance of portfolio will be minimized effectively. The model is

a minimization problem under 2 constraints which is quadratic programming model mainly.

Minimize

( )= + +


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Constraint 1

Constraint 2( ) ( ) ( )

Wolframalpha web site is used for the solution of problem under constraints.1

Table-1. Outputs of solution of the problem

Function

Domain

Global minimum ;Not founded.

Local minimum ;

Situation 1

{

|

( )

( )

Situation 2

{

|

( )

( )

Situation 3

{

|

( )

( )

6. DISCUSSIONS AND COMMENTS

According to the results the risk is minimum for the situation 1. When the situation 1 is

analyzed because the requirement (constraints of problem) of percent of each sector in portfolio

should be greater than 0 cannot be provided there is no solution. For the situation 2; percentage of

wholesale sector in portfolio should be 22, textile sector should be 2%, and construction sector

1http://www.wolframalpha.com/

http://www.wolframalpha.com/


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should be 76% in the portfolio.When we apply the outputs into the data of November 2009 the total

credits of 98.127.182,80 TL should be distributed as 21.587.980,20 TL for wholesale and retail

sector, 1.962.543,70 TL for textile sector and 74.576.658,90 TL for instruction industry. When we

apply the November 2009 sectoral performing rates to the portfolio group which is shown at

satiation 2 total performing loans will 93.642.770,54 TL. In accordance with the performing loans

91.922.762,90 TL for November 2009 the performing rate was increased from 93,6% to 95,4%.

For situation 3 the performing rate is 95,3%. When situation 2 and 3 is analyzed variance of

portfolio is and accordingly. In the theory the risk is determined

by variance so situation 2 will be selected where the risk is minimum. In other words, for both

situation 2 and 3, there is a solution, is it observed that the performing rates increased and the risk

rates decreased. The situation where the x is minimum is selected. As it is seen at the application

portfolio theory is an instrument which decrease the risks besides measuring the risks. However,

distribution of total credits to portfolio groups according to theory is not possible because of

portfolio groups own credit volume. In an example, according to the data of November 2009 total

credits of instruction sector was 29963716,9 however in the theory 74576658,90 tl credit should be

provided which is not possible technically. Although the theory is seen as impossible to applied

because of internal credit volumes of portfolio groups, it is important that the theory showed the

most effective sectors for investment. Additionally the data are total credits which were provided

by all financial institutions in Turkey. By this view, when the theory is applied by a bank the

credits distributed to portfolio groups will not exceed the internal credit volume of each sector.

Figur-1. Repayment of Loan in Level Month (%)

In graph 1 wholesale, instruction and textile and performing rates of optimum portfolio of

these sectors in a time interval (35 months) are shown. Optimum portfolio contains 22% of

wholesale-retail sector, 2% textile sector, 76% intruction sector. The performing rate of portfolio,

which is existed by these rates, is calculated as multiplication of monthly performing rates of

wholesale-retail,textile and instruction sectors with percentages. As it is seen in graph 1the

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

rep

aym

ent

of

loan

s

Wholesale

Txtile

Construction

Portfolio


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performing rate of portfolio is greater than textile and wholesale-retail sectors. Although

performing rate of instruction sector is greater than performing rate of portfolio, when the

reluctancy of instruction sector is (Z= İNSTRUCTION Y=TEXTILE X=WHOLESALE AND

RETAIL P= PORTFOLIO) =4696 , reluctancy of portfolio is

=4155 . In other

words instruction sector is riskier than portfolio on its own.

Figur-2. Repayment Risk

(In the figure, performing percentage is showed on the vertical axis and risk is showed on the horizontal axis)

It is shown that the reluctance of the portfolio is smaller than reluctance of each sector in portfolio.

( = ,

= , = ) the values show portfolio is less riskier than

each sector separately.

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E(P)=0,9663

E(Z)=0,9698

E(Y)=0,8881

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0.84

0.86

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